Why do we use $D(x \mid y)$ and not $D(x,y)$ in conditional generative adversarial networks?
It looks like you're asking about the difference between using conditional and joint probabilities.

The joint probability $$D(x,y)$$ is the probability of x and y both happening together.

The conditional probability $$D(x | y)$$ is the probability that x happens, given that y has already happened. So, $$D(x,y) = D(y) * D(x | y)$$. 

Notice that, in a C-GAN, we have some extra information that is given, like a class label $y$. We actually don't care at all about how likely that information is to appear. We care only about how likely it is to appear with a given $x$ from the source distribution, versus how likely it is to appear with a given $z$ from the generated distribution.

If you tried to minimize the joint probabilities, you would be attempting to change something that the networks have no ability to control (the chance of $y$ appearing).